Exact integral equation for pion-pion scattering involving only physical region partial waves

We derive an exact relation for ππ scattering which yields the real parts of the ππ partial wave amplitudes al(I)(s) in the region 4m2π ⩽ s ⩽ 60m2π, is given i) the I = 0 and the I = 2 S wave scattering lengths and ii) the 1m al(I)(s) for 4m2π ⩽ s ⩽ ∞, where s denotes the centre-of-mass energy, l the angular momentum and I the isotopic spin. It also provides i) a system of integral equations to determine the al(I)(s) for 4m2π ⩽ s ⩽ 16m2π, 162π ⩽ s ⩽ ∞, and ii) necessary and sufficient conditions for crossing symmetry expressed in terms of the physical region partial wave amplitudes only.